Can Reedy cofibrations be monomorphisms?

Question

In Hirschhorn's "Model Categories and Their Localizations", section 15.7, there's the following corollary to the preceding Proposition:

$\mathbb{Corollary}$ 15.7.2 If $\mathfrak{M}$ is a cellular model category and $\mathfrak{C}$ is a Reedy category, then the cofibrations of the Reedy model category on $\mathfrak{M}^{\mathfrak{C}}$ are monomorphisms.

I think we can use this to prove the following: If we've got a Reedy model category $\mathfrak{A}$ and a category $\mathcal{B}$ in which all cofibrations are monomorphisms and weak equivalences are pointwise (i.e. just a rough way to say that it's equipped with an injective model structure) then in the category $\text{Psh}(\mathfrak{A},\text{Psh}(\mathcal{B},\textit{Ssets}))$ all Reedy cofibrations are monomorphisms.

Is there a way to have the converse work for some types of Reedy model categories? (By converse I mean all monomorphisms being Reedy cofibrations)

If I'm not mistaken, this will imply some sort of relation between model structures.

Edit: Big thanks to Simon for pointing out the vague, unexplicit nature of the question.

Answer 1

I believe what you are after is the notion of "elegant Reedy category"

This sort of things isn't true for a general Reedy category, but for an elegant one $R$ (see the link for the definition) if $\mathcal{E}$ is a Grothendieck topos (for e.g. simplicial presheaves on something) in which the cofibration are the monomorphisms, then in $\mathcal{E}^{R^{op}}$ the Reedy cofibration are exactly the monomorphism. This is explained on the nLab page linked and you'll also find reference about this there.

The last section of the nLab page gives many exemples of elegant Reedy category. In particular, Reedy categories that satisfies an analogue of the Eilenberg-Zilber lemma are elegant Reedy category (most exemple come from this).